The existence of time-delayed phases (1) is not supported by oxygen uptake kinetics data. Despite many attempts for a number of years, no convincing physiological mechanism for such behavior has been proven to exist. The reason is that these time-delayed phases are a figment of the incorrect treatment of the data and the overly simple curve fitting of the, usually, averaged data. The reported problems regarding high levels of uncertainty in TD_{2} or insufficient clarity in the drop in the pulmonary gas exchange ratio, R, defining TD_{1} are due to trying to fit time-delayed phases to data with no such features. Due to the poor data handling and curve fitting the time constants are also physiologically irrelevant.

Breath-by-breath recordings exhibit spontaneous fluctuations (18). A number of different algorithms with different assumptions are therefore used to estimate the breath-by-breath V̇o_{2}, resulting in notable differences observable throughout the whole on/off transient, most extremely so in the initial response (16). These algorithms can also affect the three-phase curve parameters estimates (9, 13). Breath-by-breath variability may have biological significance (5) as nonlinear systems such as those governing the respiratory and circulatory functions can produce signals that look like random noise but are in fact not stochastic (3, 11, 14, 15, 21). Therefore part of what is attributed to noise can contain inherent features and vital information (30). For example, in both constant and free-paced 10,000-m runs the V̇o_{2} (and HR) has a scaling exponent above 0.5, the value for white noise (4).

Noise reduction is commonly achieved via ensemble averaging the responses of multiple supposedly identical exercise bouts (17). This is only justified when the noise is Gaussian and stochastic (26) and the basic response pattern of each bout is identical, which in general is not the case (2, 20). To support this procedure (17, 20) it is often quoted as showing that the noise is white. These papers however do not provide sufficient proof of the noise's whiteness for the whole on/off transient at any intensity, as only the steady states at rest or during the last 2 min (120 s is a very short sample size) of non-slow component data are analyzed. In contrast more modern studies show that some breath-by-breath algorithms produce data with non-white noise (4, 7, 9), hence averaging several repetitions can be methodologically unjustified (9). Also due to variation in parameter values on repeated testing days it is debatable whether ensemble averaging is an accurate method (2). Parameter variability is also reported, especially in the time constants (19). Differences between bouts, when ensemble averaged, can produce features not found in the raw unaveraged time series for a single bout of exercise (30). Therefore a model that is fit to the features of averaged data is not necessarily a good model of the raw unaveraged data of a single exercise bout [in which features such as time-delayed phases cannot be observed due to the high-frequency signal oscillations (5, 23)]. A curve without time-delayed phases (22–25, 28–30) can fit the data perfectly well. If the data for a single bout of exercise is instead filtered using a low-pass filter or a moving average with sufficient high *n* (30) or a more sophisticated nonlinear curve smoothing techniques (15) then the curve obtained will provide the basic response pattern for that bout of exercise. The basic response pattern is what should be modeled, not the average, which in general is a different curve (30).

The phase 1/2 components are intertwined, complicating the TD_{1} interpretation (26). In theory, the start of phase 2 (i.e., TD_{1}) should be triggered by a fall in the pulmonary gas exchange ratio (R = V̇co_{2}/V̇o_{2}), however, “this decrease is often not sufficiently clear for this purpose and a value of at least 20 s is commonly used” (26). Many researchers try to improve the phase 2 fit by constraining the fitting window to start some time after the exercise onset (26). As there exists a high degree of interdependency in the parameters (16), arbitrarily cutting data affects all the parameter values. As a result τ2 will be dependent on the amount of data removed, making it of limited use physiologically. For the phase 1 and slow component, the best fit to the data can result in unphysiologically large values of the amplitude and/or time constant (16). It is debatable therefore whether the exponential is a good model for these phases (8, 12, 26). The determination of both the phase 2 asymptote and TD_{2} is highly uncertain and via dependency, this can dramatically affect the parameter values and confidence, possibly causing an unacceptable reduction in the τ2 confidence (26).

Slow kinetics can easily be observed to exist by inspection, what is not certain however is the existence of a time-delayed slow component, nor has a physiological mechanism been proven (26). Slow kinetics emerge from the background noise after a time period, however, crucially this does not imply the existence of a time-delayed phase (26). The slow phase gain profile and time constant(s?) also remain to be determined (26). A step-wise increment in oxygen demand after a time delay TD_{2} has recently been recognized to be unrealistic and an *n*-phase curve has been proposed instead (2, 26, 27). A more powerful approach, however, (28) numerically estimates the time dependency of the oxygen demand from the on/off transient kinetics. Mathematically speaking the *n*-phase curve (2, 27) refers to the way a smooth function is approximated using first principles of infinitesimal calculus (30). A more rigorous model therefore would consist of a smooth function (23, 25).

A single exponential rise for phase-2 has been argued against as almost identical curves can be produced using very different assumptions based on numerous compartments with either a range of τ values and the same amplitudes or the same τ but different amplitudes (6, 27). Hence doubts exist regarding the phase 2 parameters physiological relevance. Regarding all the 3/*n*-phase curves the number of parameters used is large as their values depend on the exercise intensity. Ideally in a good model these parameters should be far fewer and remain constant for all exercise intensities, hence characterizing the individual (23, 25).

In conclusion, just because a curve has good statistical fit it does not mean that this is significant if the curve is not constructed from physiologically proven principles. For example, fitting straight lines point to point would result in a perfect fit having no physiological significance. Marginal statistical improvement in the fit (i.e., by adding time delays) of an arbitrary curve also have no significance, bearing in mind the spread of the raw data in a single bout of exercise. Finally as time delays cannot be seen with any sort of clarity in raw data from a single response, and bearing in mind all of the methodological problems previously discussed and the lack of a proven physiological mechanism, we have no reason to believe such features exist. To quote (10) [see also (9)] “data reporting modifications of the gas exchange parameters in several conditions and after different experimental manipulations, should be taken with a pinch of salt.”

## GRANTS

Supported by the Programs Ramon-y-Cajal 2004 and I3 2006, MICINN, Spain.

- Copyright © 2009 the American Physiological Society